Golf Impact, asymptotics

2016 April 18
by Daniel Lakeland

Here are some nice videos of golf impacts and swings

There’s a lot you can learn from these videos. But the one thing that I want to make indisputably clear here is that how you hold the club or your body or the forces you exert on the club during impact have absolutely no discernible effect on the actual forces applied to the ball.

The second video is at 68,000 fps, and replayed at say 24 frames per second, the whole impact takes less than 1 second of video watching time, which means that the impact time is about 24/68000 = 0.00035 seconds. The ball goes from zero to say 200 mph in this time. Thanks to GNU units we can get that as 26000 gees (gravitational accelerations or 1gee = 9.8 m/s^2). A golf ball weighs about 46 grams, which means at 26000g the force on the ball is 2641 lbs or the weight of a 1200 kg mass which is about as much as a typical sedan automobile. Can you lift up your car off the ground with your hands? Do you experience anything even remotely close to that force during a golf swing? No.

The speed of shear waves in steel is around 6000 m/s, a golf club is about 1 m long, so it takes 1m/(6000m/s) = .0002 s for a shear wave to travel the length of the club, so in the .0004 seconds of contact it’s just about enough time for a shear wave to travel up the club to your hand and back down around the time the ball leaves the face.

NOTHING about your hands directly affects the ball, EXCEPT position. And obviously, the way that position effects the ball is that it puts the club head at the right place to impact the ball.

Of course, leading up to impact, what you do with your hands and arms and body is critical, critical to get the club head positioned in the right place and traveling at the right speed, but during impact, all the effect comes from the club interacting with the ball as if your hands weren’t even attached.

Golf is a game of precision more than power. You need to slot that club head into the right position plus or minus less than a cm while getting it traveling at 100 mph or so. Kind of an amazing feat of control.

How does Tiger Woods get a high club speed? See the second video between 1 and 4 seconds, at the point where the club is horizontal about at the level of his right knee, his wrists begin to hinge. The club goes from an L shape at almost 90 degrees to his arms, into a straight line with his left arm by the time the club passes the ball by say 5 inches. This means the club is basically rotating around the contact point with his hand, whereas it started out rotating around a point about at his left shoulder. The radius of rotation of the club becomes about half what it was and equal about to the length of the club. Like an ice skater pulling in her arms to speed up her spin, conservation of angular momentum ensures that when the radius of rotation becomes smaller, the velocity of the club increases. During the initial downswing, his hands and wrists apply a torque to the club, this is what golfers call “the feeling of lag”. At the moment when the club is horizontal or so, he stops applying that torque, stops holding his wrist stiff, and that’s when he begins to apply only a centripetal force along the shaft to keep the club head going in a circle around his wrist. The angular momentum is $$mvR$$ where R is the radius from his shoulder pivot point to the club head, and if suddenly we reduce the radius to $$r \approx R/2$$, then $$V \approx 2v$$, essentially doubling the club speed by this slingshot effect

The above stuff about the change in radius is wrong. Yes the radius of rotation changes, but we can only invoke conservation of angular momentum if we keep the reference pivot point constant, which would be the point close to his left shoulder or something. If you have a ball on a string going around a pivot point, and you reel the string in, you will speed up the ball, but that’s not what’s going on here.

“Lagging” the club is all about applying torque so that you can get the club rotating, and then you release that stiff wrist to increase the club speed dramatically. The feeling of “retaining your lag” is about releasing the club late enough that it swings through the ball before it reaches the straight position with your left (leading) arm.

Mechanics… it works!

3 Responses
  1. Richard Kennaway permalink
    April 19, 2016

    Agreed about everything except the angular momentum. The club head doesn’t speed up when the centre of rotation of the club switches from shoulders to wrists. (Where would the energy come from?) The angular momentum changes, because now you’re measuring it relative to a different centre. Conservation does not apply. The angular velocity of the club about the wrists is twice what it was about the shoulders (for purely geometric reasons), which may create a visual illusion that the club head is moving faster, but its linear velocity does not change.

    • Daniel Lakeland
      April 19, 2016

      The issue is subtle, and I maybe shouldn’t have just shoved it in at the end. I’ll do a whole blog post on it.

      You’re right that you can’t just instantaneously change the club speed, and like a comet slingshotting around the sun, you need a source of energy. Just like the gravitational potential, which exerts a force towards the sun, provides the energy for the comet, I believe that forces exerted by the left arm are primarily responsible for the energy here. He is in essence “dragging” the club with his left arm. I’ll do a more careful analysis in a second post though.

    • Daniel Lakeland
      April 19, 2016

      Also, a rigid body has a center of rotation, you can’t just choose it. In fact, there’s a unique point where $$v = \omega \times r$$ and this point changes throughout the motion. You can construct it by looking at two points on the club, drawing their velocity vectors, and then drawing a perpendicular line to the velocity vectors and looking for the point where the two lines intersect. At any given point you can describe it as “purely rotating” about that point r, but because things aren’t just purely rotating about a fixed point, the r changes with time. In this case, we’re approximating where the position is. It doesn’t just jump from his left shoulder to his wrist, but it does move rapidly and he is exerting some significant forces to make it happen. I’m not quite sure where or how it happens at this point, until I do a more thorough analysis.

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